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Question : What is $\tan \frac{\theta}{2}$?
Option 1: $\frac{\cos \theta}{1-\sin \theta}$
Option 2: $\frac{\sin \theta}{1-\cos \theta}$
Option 3: $\frac{\cos \theta}{1-\cos \theta}$
Option 4: $\frac{\sin \theta}{1+\cos \theta}$
Correct Answer: $\frac{\sin \theta}{1+\cos \theta}$
Solution : Given: $\tan\frac{\theta}{2}$ = $\frac{\sin\frac{\theta}{2}}{\cos\frac{\theta}{2}}$ = $\frac{2\sin\frac{\theta}{2}\cos\frac{\theta}{2}}{2\cos^2\frac{\theta}{2}}$ = $\frac{2\sin\frac{\theta}{2}\cos\frac{\theta}{2}}{2\cos^2\frac{\theta}{2}-1 + 1 }$ = $\frac{\sin\theta}{\cos\theta + 1}$ Hence, the correct answer is $\frac{\sin \theta}{\cos\theta + 1}$.
Question : Directions: Six friends are sitting in a circle and facing the centre of the circle. Runa, Charu, and Pari are females. Varun, Manu and Prakash are males. Manu is between Varun and Prakash, and Charu is between Pari and Runa. Varun and Pari are opposite to each other. The person sitting to the right of Runa is male. Who is sitting immediately right to Prakash?
Option 1: Manu
Option 2: Charu
Option 3: Pari
Option 4: Varun
Correct Answer: Pari
Solution : Given: (I) Runa, Charu, and Pari are females. Varun, Manu and Prakash are males. Charu is between Pari and Runa. Varun and Pari are opposite to each other. The person sitting to the right of Runa is male.
(II) Manu is between Varun and Prakash.
Question : A diagonal relationship is found between which pair of the following elements?
Option 1: Li and Mg
Option 2: Be and B
Option 3: Be and Mg
Option 4: Na and Mg
Correct Answer: Li and Mg
Solution : The correct answer is Li and Mg.
The relationship exists between Li (Lithium) and Mg (Magnesium). This means that they are in different parts of the periodic table, they share some similar properties because their atoms and ions are roughly the same
Question : If in a right-angled $\triangle P Q R$, $\tan Q=\frac{5}{12}$, then what is the value of $\cos Q$?
Option 1: $\frac{12}{5}$
Option 2: $\frac{13}{5}$
Option 3: $\frac{12}{13}$
Option 4: $\frac{5}{13}$
Correct Answer: $\frac{12}{13}$
Solution : Given, $\tan Q = \frac{5}{12}$ We know, $\tan\theta = \frac{\text{Perpendicular}}{\text{Base}}$ Applying Pythagoras theorem, $\small\text{Hypotenuse}^2 = \text{Perpendicular}^2+\text{Base}^2$ ⇒ $\text{H}^2=5^2+12^2$ ⇒ $\text{H}^2 = 25 + 144$ ⇒ $\text{H}=\sqrt{169}$ ⇒ $\text{H} = 13\ \text{units}$ Now, $\cos\theta = \frac{\text{Base}}{\text{Hypotenuse}}$ ⇒ $\cos Q = \frac{12}{13}$ Hence, the correct answer is
Question : Which of the following is a semiconductor?
Option 1: Silver
Option 2: Copper
Option 3: Glass
Option 4: Silicon
Correct Answer: Silicon
Solution : The correct option is Silicon.
Silicon is a widely used semiconductor material in electronics. It possesses electrical conductivity levels situated between those of conductors (such as metals) and insulators (like rubber or glass). Silicon exhibits properties that render it exceptionally suitable for crafting electronic
Question : A milkman mixes water with milk and sells the mixture at the cost price of pure milk. The volume of water in litres to be mixed with each litre of milk to get a 25% profit is:
Option 1: $\frac{1}{4}$ litre
Option 2: $\frac{1}{5}$ litre
Option 3: $1\frac{1}{4}$ litre
Option 4: Cannot be calculated without knowing the cost price of milk.
Correct Answer: $\frac{1}{5}$ litre
Solution : Let the cost price (CP) of 1 litre of milk be Re. 1. So, the selling price (SP) of 1 litre of the mixture is Re. 1. To get a 25% profit, the CP of 1 litre of the mixture = $\frac{100}{125}$ × 1
Question : C1, C2 and C3 can do work alone in 10, 12, and 15 days respectively. All three of them began the work together but C2 left 2 days before the completion of the work. In how many days was the work completed?
Option 1: 3
Option 2: 7
Option 3: 5
Option 4: 4
Correct Answer: 5
Solution : C1, C2 and C3 can do a work alone in 10,12 and 15 days, respectively. C1's 1 day's work = $\frac{1}{10}$ C2's 1 day's work = $\frac{1}{12}$ C3's 1 day's work = $\frac{1}{15}$ (C1 + C2 + C3)'s 1 day's work = $\frac{1}{10}+\frac{1}{12}+\frac{1}{15}=\frac{1}{4}$ So, they
Question : Article 60 of the Indian Constitution refers to:
Option 1: election of the President of India
Option 2: oath of the president
Option 3: impeachment of the president
Option 4: Vice President of India
Correct Answer: oath of the president
Solution : The correct option is the oath of the president.
Article 60 of the Indian Constitution deals with the "oath of office by the president". It outlines the oath or affirmation that the president takes before entering the office. Article 54 of
Question : Directions: In the following question a series is given, with one number missing. Choose the correct alternative from the given ones that will complete the series. 5, 11, 17, 25, 33, 43, ?
Option 1: 49
Option 2: 51
Option 3: 52
Option 4: 53
Correct Answer: 53
Solution : Given: 5, 11, 17, 25, 33, 43, ?
Add the consecutive even numbers starting from 6 to every two terms – 5 + 6 = 11; 11 + 6 = 17; 17 + 8 = 25; 25 + 8 = 33; 33 + 10 =
Question : Directions: Three statements are given, followed by three conclusions numbered I, II, and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements. Statements: No toy is beautiful. Some beautiful are actresses. All actresses are rich. Conclusions: I. No toy is rich. II. Some actresses are beautiful. III. Some actresses are rich.
Option 1: Only conclusions II and III follow
Option 2: Only conclusions I and III follow
Option 3: All the conclusions follow
Option 4: Only conclusions I and II follow
Correct Answer: Only conclusions II and III follow
Solution : The possible Venn diagram according to the given statements is as follows –
Let's analyse the conclusions – Conclusion (I): No toy is rich – From the Venn diagram, it is evident that there is no direct relation between toy
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